56a PHILOSOPHICAL TRANSACTIONS. [aNNO 1748. 



able to trace its relations through a long process of reasoning, and with a perspi- 

 cuity and accuracy which we in vain expect in subjects not capable of mensu- 

 ration. 



Extended quantities, such as lines, surfaces and solids, besides what they have 

 in common with all other quantities, have this peculiar, that their parts have a 

 particular place and disposition among themselves : a line may not only bear any 

 assignable proportion to another, in length or magnitude, but lines of the same 

 length may vary in the disposition of their parts; one may be straight, another 

 may be part of a curve of any kind or dimension, of which there is an endless 

 variety. The like may be said of surfaces and solids. So that extended quan- 

 tities admit of no less variety with regard to their form, than with regard to their 

 magnitude : and as their various forms may be exactly defined and measured, no 

 less than their magnitudes, henee it is that geometry, which treats of extended 

 quantity, leads us into a much greater compass and variety of reasoning than 

 any other branch of mathematics. Long deductions in algebra for the most part 

 are made, not so much by a train of reasoning in the mind, as by an artificial 

 kind of operation, which is built on a few very simple principles : but in geome- 

 try we may build one proposition on another, a third upon that, and so on, with- 

 out ever coming to a limit which we cannot exceed. The properties of the 

 more simple figures can hardly be exhausted, much less those of the more 

 complex ones. 



It may be deduced from what has been said above, that mathematical evi- 

 dence is an evidence sui generis, not competent to any proposition which does 

 not express a relation of things measurable by lines or numbers. All proper 

 quantity may be measured by these, and improper quantities must be measured 

 by those that are proper. 



There are many things capable of more and less, which perhaps are not capable 

 of mensuration. Tastes, smells, the sensations of heat and cold, beauty, plea- 

 sure, all the affections and appetites of the mind, wisdom, folly, and most kinds 

 of probability, with many other things too tedious to enumerate, admit of de- 

 grees, but have not yet been reduced to measure, nor perhaps ever can be. I 

 say, most kinds of probability, because one kind of it, viz. the probability of 

 chances, is properly measurable by number, as observed above. 



Though attempts have been made to apply mathematical reasoning to some of 

 these things, and the quantity of virtue and merit in actions has been measured 

 by simple and compound ratios ; yet Dr. M. does not think that any real know- 

 ledge has been struck out this way : it may perhaps, if discreetly used, be a help 

 to discourse on these subjects, by pleasing the imagination, and illustrating what 

 is already known ; but till our affections and appetites shall themselves be reduced 

 to quantity, and exact measures of their various degrees be assigned, in vain 



